R ¥ RYdR y

¬§ V§ A¨

©²

XR

3. Solve 3 H

©¦¬ ©

RX`R w

4. Set

’UQH

W

H

5. EndDo

6. EndDo

µ£ „ ¢

|5¥ j qz A C¡y 5

„ 5| | 5£§|

¢ ¡

£ ¡C

@¡

¨§ £ "!

As was the case with the scalar algorithms, there is only a slight difference between

the Jacobi and Gauss-Seidel iterations. Gauss-Seidel immediately updates the component

to be corrected at step , and uses the updated approximate solution to compute the residual

E

vector needed to correct the next component. However, the Jacobi iteration uses the same

©

previous approximation for this purpose. Therefore, the block Gauss-Seidel iteration

H

can be de¬ned algorithmically as follows:

• c

‘ • —R˜ “f

˜Q¤ v ¢

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5 ¥Q 5

4E D CA479( R¤B9

B @8 5 6 5

B

G 6 QS 8 5S

3 £A

' 7$

%# ) 2

1 ¢¢

1. Until convergence Do:

T

¬vE

2. For Do: (˜ (

( ¢

¡

X

R ¥

R X dR y

§ ¬ V§ A¨

©²

R

3. Solve 3

©¬©

R XR w

4. Set

5. EndDo

6. EndDo

From the point of view of storage, Gauss-Seidel is more economical because the new ap-

proximation can be overwritten over the same vector. Also, it typically converges faster. On

the other hand, the Jacobi iteration has some appeal on parallel computers since the second

Do loop, corresponding to the different blocks, can be executed in parallel. Although the

¡

point Jacobi algorithm by itself is rarely a successful technique for real-life problems, its

block Jacobi variant, when using large enough overlapping blocks, can be quite attractive

especially in a parallel computing environment.

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§ ! E B ©' 0A ©' ) 63 ©C8 ¡5 ) B 3 A 8 §©DC7GP5 A B

I E 'B A 8 3

EF

5

E B B©E

The Jacobi and Gauss-Seidel iterations are of the form

r± w°i ¯®

© ¦ ¬

© ( w

"VH

WU H

in which T

r± w°i ¯®

§ W “ T&² ¬ XX § T ¦

# (

±r 0 i ¯®

X

§ W “ &² # ² ¬ § 0 ( ¦

% (

for the Jacobi and Gauss-Seidel iterations, respectively. Moreover, given the matrix split-

ting

p±q0 i ¯®

°

¬§ ² ”

(

§

where is associated with the linear system (4.1), a linear ¬xed-point iteration can be

de¬ned by the recurrence

r±0 0 i ¯®

w

© ¬ © ¨

W “ ( W“

"QH

WU H

£ q¦ ¡k q`¤¨¤¦tst ¢ V£¤

s©| | §£¥ j ¥ ¡C

£¡¨ !p

§

¢

which has the form (4.18) with T

a±a0 i ¯®

§ W “ ² ¬ X ² W “ ¬ W “ ¬ ¦

§ ¬ ¨

( –W “

# ² § ( # ¬

¬

For example, for the Jacobi iteration, , while for the Gauss-Seidel

§ w ¬ §¬ ( % #`¬

²©

' ¬ ²

iteration, .

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The iteration can be viewed as a technique for solving the system

T

W

H

X

¬ © ¦ ²

§ W “ ² ¬ ¦